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B3.1 Quantum Structural Methods for Atoms and Molecules Quantum structural methods for atoms and molecules 1907 B3.1 Quantum structural methods for atoms and molecules Jack Simons B3.1.1 What does quantum chemistry try to do? Electronic structure theory describes the motions of the electrons and produces energy surfaces and wave- functions. The shapes and geometries of molecules, their electronic, vibrational and rotational energy levels, as well as the interactions of these states with electromagnetic fields lie within the realm of quantum structure theory. B3.1.1.1 The underlying theoretical basis—the Born–Oppenheimer model In the Born–Oppenheimer [1] model, it is assumed that the electrons move so quickly that they can adjust their motions essentially instantaneously with respect to any movements of the heavier and slower atomic nuclei. In typical molecules, the valence electrons orbit about the nuclei about once every 10−15 s (the inner-shell electrons move even faster), while the bonds vibrate every 10−14 s, and the molecule rotates approximately every 10−12 s. So, for typical molecules, the fundamental assumption of the Born–Oppenheimer model is valid, but for loosely held (e.g. Rydberg) electrons and in cases where nuclear motion is strongly coupled to electronic motions (e.g. when Jahn–Teller effects are present) it is expected to break down. This separation-of-time-scales assumption allows the electrons to be described by electronic wavefunc- tions that smoothly ‘ride’ the molecule’s atomic framework. These electronic functions are found by solving ˆ a Schrodinger¨ equation whose Hamiltonian He contains the kinetic energy Te of the electrons, the Coulomb repulsions among all the molecule’s electrons Vee, the Coulomb attractions Ven among the electrons and all of the molecule’s nuclei, treated with these nuclei held clamped, and the Coulomb repulsions Vnn among all of these nuclei, but it does not contain the kinetic energy TN of all the nuclei. That is, this Hamiltonian keeps the nuclei held fixed in space. The electronic wavefunctions ψk and energies Ek that result ˆ Heψk = Ekψk thus depend on the locations {Qi } at which the nuclei are sitting. That is, the Ek and ψk are parametric functions of the coordinates of the nuclei, and, of course, the wavefunctions ψk depend on the coordinates of all of the electrons. These electronic energies’ dependence on the positions of the atomic centres cause them to be referred to as electronic energy surfaces such as that depicted below in figure B3.1.1 for a diatomic molecule. For nonlinear polyatomic molecules having N atoms, the energy surfaces depend on 3N − 6 internal coordinates and thus can be very difficult to visualize. In figure B3.1.2, a ‘slice’ through such a surface is shown as a function of two of the 3N − 6 internal coordinates. The Born–Oppenheimer theory is soundly based in that it can be derived from a Schrodinger¨ equation describing the kinetic energies of all electrons and of all N nuclei plus the Coulomb potential energies of 1908 Quantum structural methods for atoms and molecules 4 2 0 -2 Energy -4 -6 01234 Internuclear distance Figure B3.1.1. Energy as a function of internuclear distance for a typical bound diatomic molecule or ion. Figure B3.1.2. Two-dimensional slice through a (3N − 6)-dimensional energy surface of a polyatomic molecule or ion. After [2]. What does quantum chemistry try to do? 1909 interaction among all electrons and nuclei. By expanding the wavefunction that is an eigenfunction of this full Schrodinger¨ equation in the complete set of functions {ψk} and then neglecting all terms that involve derivatives of any ψk with respect to the nuclear positions {Qi }, one can separate variables such that: (1) the electronic wavefunctions and energies obey ˆ Heψk = Ekψk (2) the nuclear motion (i.e. vibration/rotation) wavefunctions obey ˆ (TN + Ek)χk,L = Ek,Lχk,L where TN is the kinetic energy operator for movement of all nuclei. Each and every electronic energy state, labelled k, has a set, labelled L, of vibration/rotation energy levels Ek,L and wavefunctions χk,L. B3.1.1.2 Non-Born–Oppenheimer corrections—radiationless transitions Because the Born–Oppenheimer model is obtained from the full Schrodinger¨ equation by making approxi- mations, it is not exact. Thus, in certain circumstances it becomes necessary to correct the predictions of the Born–Oppenheimer theory (i.e. by including the effects of the neglected coupling terms using perturbation theory). For example, when developing a theoretical model to interpret the rate at which electrons are ejected from rotationally/vibrationally hot NH− ions, we had to consider [3] coupling between: (1) 2 NH− in its v = 1 vibrational level and in a high rotational level (e.g. J>30) prepared by laser excitation of vibrationally ‘cold’ NH− in v = 0 having high J (due to natural Boltzmann populations), see figure B3.1.3; and (2) 3− NH neutral plus an ejected electron in which the NH is in its v = 0 vibrational level (no higher level is energetically accessible) and in various rotational levels (labelled N). Because NH has an electron affinity of 0.4 eV, the total energies of the above two states can be equal only if the kinetic energy KE carried away by the ejected electron obeys − KE = Evib/rot(NH (v = 1,J))− Evib/rot(NH (v = 0,N))− 0.4eV. In the absence of any coupling terms, no electron detachment would occur. It is only by the anion converting some of its vibration/rotation energy and angular momentum into electronic energy that the electron that − occupies a bound N2p orbital in NH can gain enough energy to be ejected. My own research efforts [4] have, for many years, involved taking into account such non-Born–Oppen- heimer couplings, especially in cases where vibration/rotation energy transferred to electronic motions causes electron detachment, as in the NH− case detailed above. Professor Yngve Ohrn¨ has been active [5] in attempting to avoid using the Born–Oppenheimer approximation and, instead, treating the dynamical motions of the nuclei and electrons simultaneously. Professor David Yarkony has contributed much [6] to the recent treatment of non-Born–Oppenheimer effects and to the inclusion of spin–orbit coupling in such studies. B3.1.1.3 What is learned from an electronic structure calculation? The knowledge gained via structure theory is great. The electronic energies Ek(Q) allow one to determine [7] the geometries and relative energies of various isomers that a molecule can assume by finding those geometries {Qi } at which the energy surface Ek has minima ∂Ek/∂Qi = 0, with all directions having positive 2 curvature (this is monitored by considering the so-called Hessian matrix Hi,j = ∂ Ek/∂Qi ∂Qj : if none of its 1910 Quantum structural methods for atoms and molecules NH - (v=1; J) x NH (v'=0; N) + e - NH - (v=0; J") Figure B3.1.3. Energies of NH− and of NH pertinent to the autodetachment of v = 1, J levels of NH− formed by laser excitation of v = 0, J NH−. eigenvalues are negative, all directions have positive curvature). Such geometries describe stable isomers, and the energy at each such isomer geometry gives the relative energy of that isomer. Professor Berny Schlegel [8] has been one of the leading figures in using gradient and Hessian information to locate stable structures and transition states. Professor Peter Pulay [9] has done as much as anyone to develop the theory that allows us to compute gradients and Hessians for most commonly used electronic structure methods. There may be other geometries on the Ek energy surface at which all ‘slopes’ vanish ∂Ek/∂Qi = 0, but at which not all directions possess positive curvature. If the Hessian matrix has only one negative eigenvalue, there is only one direction leading downhill away from the point {Qi } of zero force; all the remaining directions lead uphill from this point. Such a geometry describes that of a transition state, and its energy plays a central role in determining the rates of reactions which pass through this transition state. The energy surface shown in figure B3.1.2 displays such transition states, and it also shows a second-order saddle point (i.e. a point where the gradient vanishes and the Hessian has two directions of negative curvature). At any geometry {Qi }, the gradient vector having components ∂Ek/∂Qi provides the forces (Fi = −∂Ek/∂Qi ) along each of the coordinates Qi . These forces are used in molecular dynamics simulations which solve the Newton F = ma equations and in molecular mechanics studies which are aimed at locating those geometries where the F vector vanishes (i.e. the stable isomers and transition states discussed above). Also produced in electronic structure simulations are the electronic wavefunctions {ψk} and energies {Ek} of each of the electronic states. The separation in energies can be used to make predictions on the spectroscopy of the system. The wavefunctions can be used to evaluate the properties of the system that depend on the spatial distribution of the electrons. For example, the z component of the dipole moment [10] of a molecule µz can be computed by integrating the probability density for finding an electron at position r multiplied by the ∗ z coordinate of the electron and the electron’s charge e: µz = eψk ψkz dr. The average kinetic energy of ∗ 2 2 an electron can also be computed by carrying out such an average-value integral: ψk (−h¯ /2me∇ )ψk dr. The rules for computing the average value of any physical observable are developed and illustrated in popular undergraduate text books on physical chemistry [11] and in graduate-level texts [12].
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